How Dangerous Is Vioxx?

Vioxx has been removed from the market because it has been linked to an increased chance of cardiac events. But, how much extra risk is there? That sort of information is harder to come by.

Here is what I constructed from the information in a September 29 article entitled "Coping Without Vioxx" in the Wall Street Journal.

In the study that produced the problem, there were 2,600 participants. Some got a placebo. They're rate of cardiac events was 7.5 per 1000. The rate among people who got Vioxx was 15 per 1000. Does this make Vioxx a problem?

Most users of statistics never grasp the idea that there are two ways to ask this question, but that you can't use both - you must choose one or the other.

Here's the skinny. Can we be sure (from this data) that Vioxx leads to extra cardiac events? Yes (and we can be well over 99% sure of that).

But how many more cardiac events you may well ask? That's the second question, and it really isn't methodologically fair to ask this, after the first one has already been addressed. But people do anyway, so let me perpetuate the problem by answering it. The problem with a figure like 7.5 out of 1000 is that it is an average - no group of 1000 people will ever suffer 7.5 events. It's kind of like saying you football team averages 22.2 points per game - that may be useful in some contexts, but they'll never score that many points exactly. A bigger problem is that even if the rate is 7.5 per 1000, does that mean that next time there might not be 8 or 5 or 11 people that have events? Unfortunately, it doens't. So, what exactly can statistics say in this case? What is can do is put a range on how many events there should be, and attach a certainty to that statement. From the data in this article we can be pretty certain (over 95% certain) that people on the placebo will suffer between 2 and 13 events per 1000. Turning this around means that we are pretty certain that no fewer than 2 out of 1000 people on Vioxx (and perhaps several more) will have cardiac events solely due to using Vioxx.

A simple application of the normal approximation to the binomial distribution can give you answers to these problems (if you took one semester of statistics, you learned this once upon a time).

I have to make one assumption. I don't know exactly how many people got the placebo, so I'm going to say half. You then figure out what 7.5 out of a 1000 works out to be when you actually are working with 1300 people. It rounds to 10 events out of 1300 (and 1,290 non-events). The calculation is then the probability of an event (10 divided by 1300), times the probability of a non-event (1,290 out of 1300), divided by the number of people (1300). Take the square root of that and multiply by the number of people (1300) and you get the standard error for the number of people out of 1300 who should have cardiac events. This is just over 3.

For the first question, about 20 people had events on Vioxx, and about 10 had them on the placebo. The standard error says that we can be sure that the difference between those numbers (10) is very unlikely to have occurred by chance (so that it must have occurred because there is something bad with Vioxx) because 20 is more than 3 standard errors above 10. Statistical tables, or a spreadsheet would put the odds on that as around 0.1%.

Alternatively, you can say that it is pretty likely that if Vioxx does not cause harm, then the number of people who have events while using it should be 10 plus or minus 2 times that standard error of 3 (out of 1300). So, we'd be pretty sure that Vioxx was OK if there were between 4 and 16 cardiac events among 1300 users. And there appear to be a few extra cases ... and Vioxx is the source.

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